This probably varies greatly, especially when you look at the different time periods, but are their traditions on raising a royal? Are/ were they raised away from court by servants, and tutors? If so, were they traditionally brought to court at a certain age? Like a debutante ball or male equivalent? Did/do they get their own seals/heraldic symbols etc at a certain age? Did they wear or carry something to identify them as a royal?
I think this would vary in different cultures as well as different time periods. Are you writing a novel? You might want to start by picking a particular country and era -- that might help you with your research. Certainly in medieval England, royal children would at least sometimes be raised away from their parents, with their own households. I can't think of a specific book on that topic, but if you were interested in, say, the Tudor era, you could read about how Henry VIII and Elizabeth I etc. were raised to help answer some of your questions.